Slope-Intercept Form | Linear Equations

Exercises 1 to 5: Basic Applications

1 Finding slope and y-intercept

Exercise 1

Identify the slope and y-intercept of the equation:
$y = 3x + 7$

Definition:

Slope-intercept form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept

Method:

Compare the given equation with the standard form $y = mx + b$
Identify the coefficient of $x$ as the slope ($m$)
Identify the constant term as the y-intercept ($b$)

Standard form

$y = mx + b$

Given equation

$y = 3x + 7$

Step 1: Compare with standard form

$y = mx + b$ vs $y = 3x + 7$

Step 2: Identify the slope

The coefficient of $x$ is $3$, so slope $m = 3$

Step 3: Identify the y-intercept

The constant term is $7$, so y-intercept $b = 7$

Slope: $m = 3$, Y-intercept: $b = 7$

Final answer:

Slope = 3, Y-intercept = 7

Applied rules:

• Standard form: $y = mx + b$ identifies slope and y-intercept directly

• Coefficient identification: The number multiplying $x$ is the slope

• Constant term: The standalone number is the y-intercept

2 Writing equation from slope and y-intercept

Exercise 2

Write the equation of a line with slope = -2 and y-intercept = 5

Definition:

Slope-intercept form: $y = mx + b$ where $m$ is slope and $b$ is y-intercept

Given values

$m = -2$, $b = 5$

Substitute in form

$y = mx + b$

Final equation

$y = -2x + 5$

Step 1: Write the standard form

$y = mx + b$

Step 2: Substitute the given values

$y = (-2)x + (5)$

Step 3: Simplify the equation

$y = -2x + 5$

Final answer:

$y = -2x + 5$

Applied rules:

• Direct substitution: Replace $m$ and $b$ with given values

• Sign preservation: Negative slopes remain negative

• Form consistency: Maintain the standard form structure

3 Converting to slope-intercept form

Exercise 3

Convert to slope-intercept form: $2x + 3y = 12$

Definition:

Slope-intercept form: $y = mx + b$ with $y$ isolated on one side

Original equation

$2x + 3y = 12$

Isolate y-term

$3y = -2x + 12$

Divide by coefficient

$y = -\frac{2}{3}x + 4$

Step 1: Move x-term to the right side

$2x + 3y = 12$ → $3y = -2x + 12$

Step 2: Divide every term by the y-coefficient

$3y = -2x + 12$ → $y = -\frac{2}{3}x + 4$

Step 3: Identify slope and y-intercept

Slope: $m = -\frac{2}{3}$, Y-intercept: $b = 4$

$y = -\frac{2}{3}x + 4$

Final answer:

$y = -\frac{2}{3}x + 4$

Applied rules:

• Isolation: Get $y$ alone on one side of the equation

• Division: Divide ALL terms by the coefficient of $y$

• Sign handling: Be careful with negative signs during rearrangement

Exercises 6 to 10: Advanced Applications

6 Word problem application

Exercise 6

A taxi charges $3 per mile plus a $2 base fare. Write an equation for the total cost.

Definition:

Linear relationship: Cost = Rate × Distance + Fixed amount

Variables defined

Let $x$ = miles, $y$ = cost

Rate and base

Slope = 3, Intercept = 2

Final equation

$y = 3x + 2$

Step 1: Define variables

Let $x$ = number of miles traveled, $y$ = total cost

Step 2: Identify rate and base

Rate per mile = $3 (slope), Base fare = $2 (y-intercept)

Step 3: Write the equation

$y = 3x + 2$

Final answer:

$y = 3x + 2$ where $x$ is miles and $y$ is cost in dollars

Applied rules:

• Variable definition: Clearly define what each variable represents

• Rate identification: The rate of change is the slope

• Fixed amount: The starting value is the y-intercept

7 Finding y-value given x-value

Exercise 7

If $y = -\frac{1}{2}x + 6$, find $y$ when $x = 4$

Definition:

Function evaluation: Substitute the given $x$-value into the equation

Original equation

$y = -\frac{1}{2}x + 6$

Substitute x = 4

$y = -\frac{1}{2}(4) + 6$

Calculate result

$y = 4$

Step 1: Substitute the given value

$y = -\frac{1}{2}(4) + 6$

Step 2: Perform the multiplication

$-\frac{1}{2} \times 4 = -2$

Step 3: Complete the calculation

$y = -2 + 6 = 4$

$y = 4$ when $x = 4$

Final answer:

When $x = 4$, $y = 4$

Applied rules:

• Substitution: Replace $x$ with the given value

• Order of operations: Perform multiplication before addition

• Fraction arithmetic: Multiply numerators and denominators correctly

Visual Learning: Slope-Intercept Form

Slope-Intercept Form

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Definition

y = mx + b

m = slope (rise over run)

b = y-intercept (where line crosses y-axis)

Slope Examples

m > 0

↗️ Positive

m < 0

↘️ Negative

m = 0

➡️ Horizontal

Y-Intercept Examples

b > 0

⬆️ Above origin

b < 0

⬇️ Below origin

b = 0

🎯 Through origin

Identify m and b

Plot y-intercept

Use slope to plot

Draw the line

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Key Point 1: Slope tells direction and steepness

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Key Point 2: Y-intercept is the starting point

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Key Point 3: Always solve for y first

Slope-Intercept Form: Complete Guide with Exercises and Solutions

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